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- Show that if
equal spheres of water coalesce so as to form a single spherical drop then the
surface energy is decreased by a factor
. (Lamb 1928.)

- A circular cylinder of radius
, height
, and specific gravity
floats upright in water. Show
that the depth of the base below the general level of the water surface is

where
is the surface tension at the air/water interface, and
the contact angle of the
interface with the cylinder. (Lamb 1928.)

- A film of water is held between two parallel plates of glass a small distance
apart. Prove
that the apparent attraction between the plates is

where
is the surface tension at the air/water interface,
the angle of contact of the
interface with glass,
the area of the film, and
the circumference of the film. (Lamb 1928.)

- Show that if the surface of a sheet of water is slightly corrugated then the surface energy is
increased by

per unit breadth of the corrugations. Here,
is measured horizontally, perpendicular to the corrugations.
Moreover,
denotes the elevation of the surface above the mean level. Finally,
is the
surface tension at an air/water interface. If the corrugations are sinusoidal, such that

show that the average increment of the surface energy per unit area is

(Lamb 1928.)

- A mass of liquid, which is held together by surface tension alone, revolves about a
fixed axis at a small angular velocity
, so as to assume a slightly spheroidal shape of
mean radius
. Prove that the ellipticity of the spheroid is

where
is the uniform mass density, and
the surface tension. [If
is the maximum radius, and
the minimum radius, then
, and
the ellipticity is defined
.] (Lamb 1928.)

- A liquid mass rotates, in the form of a circular ring of radius
and small cross-section, with a constant angular
velocity
, about an axis normal to the plane of the ring, and passing through its center. The mass is
held together by surface tension alone. Show that the section of the ring must be approximately circular.
Demonstrate that

where
is the density,
the surface tension, and
the radius of the cross-section. (Lamb 1928.)

- Two spherical soap bubbles of radii
and
are made to coalesce. Show that when the
temperature of the gas in the resulting bubble has returned to its initial value the radius
of
the bubble satisfies

where
is the ambient pressure, and
the surface tension of the soap/air interfaces. (Batchelor 2000.)

- A rigid sphere of radius
rests on a flat rigid surface, and a small amount of liquid surrounds
the contact point, making a concave-planar lens whose diameter is small compared to
. The
angle of contact of the liquid/air interface with each of the solid surfaces is zero, and the surface tension of the interface is
. Show that there is an adhesive force of magnitude
acting
on the sphere. (It is interesting to note that the force is independent of the volume of liquid.) (Batchelor 2000.)

- Two small solid bodies are floating on the surface of a liquid. Show that the effect of
surface tension is to make the objects approach one another if the liquid/air interface has either an acute or an obtuse angle
of contact with both bodies, and to make them move away from one another if the interface has an acute
angle of contact with one body, and an obtuse angle of contact with the other. (Batchelor 2000.)

** Next:** Incompressible Inviscid Flow
** Up:** Surface Tension
** Previous:** Axisymmetric Soap-Bubbles
Richard Fitzpatrick
2016-01-22